Long time existence of Smooth solution for the porous medium equation in a bounded domain

Abstract

In this paper, we are going to show the long time existence of the smooth solution for the porous medium equations in a smooth bounded domain: {equation} {cases} u_t=\La u^m\quad\text{in $\Omega\times [0,\infty)$} u(x,0)=u_0>0\quad\text{in $\Omega$} u(x,t)=0\quad\text{for $x\in\partial\Omega$} {cases} {equation} where $m>1$ is the permeability. The proof is based on the short time existence of $C^{2,\bar{\gamma}}_{s}$-smooth solution, the global $C^{1}_{s}$-estimate, the H\"older estimate of divergence type degenerate equation with measurable coefficients and $C^{1,\bar{\gamma}}_s$-estimate of mixed type equation with Lipschitz coefficients.